Andrew, your accident sounds like a very severe one. I’m glad the outcome wasn’t more severe as you’ve intimated it might have been. Above all, the best for your recovery and I hope you are able to get back to the birds soon.
I noted your earlier comment about wanting to stick with Canon. The only reason I initially suggested Nikon was that you opened the discussion up with another brand. As a Canon user myself and great believer in the power of DPP, I only suggest Nikon as I suspect they have the edge at the moment in the crop sensor cameras. Either way, I think Ian’s advice is right that a crop sensor Canon like the 80D or 7DII (or Mark III if it ever comes out) is a great choice if you want to stay in the Canon camp.
I’m less sure about your 400 f5.6. Optically it isn’t the greatest although on a full frame like your 5DIII it may be fine. The pixels on that camera are a lot larger than on the crop sensor cameras so any softness or other defects in the optics are going to be magnified. But you have it and if you go down the crop sensor route, it’s certainly worth a try. The other issue, as Ian notes, is the lack of image stabilisation. I hadn’t realised how important this is until I accidentally switched my lens stabiliser off one time. I had to discard a high proportion of the shots from that session.
As for alternative lenses, the Canon 300 f4 may be worth a look although optically it is not top of the list either. But it is relatively light and can be coupled with a 1.4x converter. That will impact image quality though. If extra weight can be considered, the best choice would be the 100-400 MkII which is optically a significant improvement on the Mark I lens and sharper than the 300 f4: I know, I had the 100-400 MkI and upgraded. I am very happy with the results I get on my crop sensor Canon 80D.